On a degenerate non-local parabolic problem describing infinite dimensional replicator dynamics

Abstract

We establish the existence of locally positive weak solutions to the homogeneous Dirichlet problem for \[ u_t = u \Delta u + u \int_\Omega |\nabla u|^2 \] in bounded domains $\Omega\subset\mathbb{R}^n$ and prove that solutions converge to $0$ if the initial mass is small, whereas they undergo blow-up in finite time if the initial mass is large. We show that in this case the blow-up set coincides with $\overline{\Omega}$, i.e. the finite-time blow-up is global. Key words: Degenerate diffusion, non-local nonlinearity, blow-up, evolutionary games, infinite dimensional replicator dynamics